Regularity of Extremal Solutions in Fourth Order Nonlinear Eigenvalue Problems on General Domains

Details Regularity of Extremal Solutions in Fourth Order Nonlinear Eigenvalue Problems on General Domains Regularity of Extremal Solutions in Fourth Order Nonlinear Eigenvalue Problems on General Domains

2010-03-19

arxiv.org/abs/1003.3862v1

Mathematics

Analysis of PDEs

19 pages. Updated versions - if any - can be downloaded at http://www.birs.ca/~nassif/

Scientific paper

We examine the regularity of the extremal solution of the nonlinear eigenvalue problem $\Delta^2 u = \lambda f(u)$ on a general bounded domain $\Omega$ in $ \IR^N$, with the Navier boundary condition $ u=\Delta u =0 $ on $ \pOm$. Here $ \lambda$ is a positive parameter and $f$ is a non-decreasing nonlinearity with $f(0)=1$. We give general pointwise bounds and energy estimates which show that for any convex and superlinear nonlinearity $f$, the extremal solution $ u^*$ is smooth provided $N\leq 5$.

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